Hyperspectral Data Processing: Algorithm Design and Analysis

10.1 Introduction

In Chapters 7 and 08, we have witnessed the inconsistency in final results produced by an EEA using random initial conditions. This dilemma can be resolved by ID-EEAs developed in Chapter 9, which use a set of specific initial endmembers generated by a custom-designed initialization algorithm. This chapter investigates an idea completely opposite to that used in ID-EEAs. It makes the disadvantage of using random initial endmembers an advantage for an EEA. Its idea originates from the concept of a random variable where realizations resulting from physical experiments conducted for a random variable constitute an ensemble of outcomes produced by the random variable. The following example will shed light on how an EEA using random initial conditions can be considered as a random algorithm.

Now consider a problem of estimating the bias of a coin, which is defined as the probability of a head turned up, θ. Then a process of flipping the coin in a fixed N trials (i.e., N times) can be considered a random algorithm where the randomness is caused by the uncertainty with the bias specified by the parameter θ. In order to estimate the value of θ, we flip the coin N times, which produces N outcomes, each of which is either a head or a tail. So, a single run resulting from such a N-coin flipping experiment produces a realization which consists of N outcomes. Then the parameter θ can be estimated from a realization by an estimator where represents N outcomes resulting from ...